Elementary Mathematical Analysis — DU B.A. (Prog) Mathematics as Major Previous Year Question Papers

Your Roll N............... 1. (a) Define the limit of a function in terms of e and 8. Use € - & definition, to show that tim (4x - 5) = 3. (7.5) (b) State the sequential criterion for limits. Use it to prove that lim f(x) does not x→X0 1, if x is rationa exist, where (x) = -1, if x is irrationa. (7.5) (e) Let f:A → 8 where A SIR and let to be a cluster point of A. Then limo f(x) = L if and only if for each sequence < Kin > in A \< Xo > that converges to xo, the sequence < f(X) > converges to L. (7.5) 2. (a) Define the continuity of a function at a point in terms of € - 8. Use € - & definition, to show that f(x) = 4x2 - 5x - 3 is continuous at Xo = 2. (7.5) (b) State the Intermediate Value Theorem. Also, prove that cos x = x for some x in (c) Define uniform continuity of a function f on interval I. Use it to prove that the function f(x) = 3x? - 2x - 1 is uniformly continuous on the interval [-1, 5]. (7.5) 3. If f: [a, b] → R is a bounded function and P and 2 are any partitions of la, b], then prove that S(, P) ≤ $(f, 2). (b) Prove that f(x) = 7 is integrable on [-1, 1]. Also, find If. (7.5) (c) Let f(x) = 5 - 3x be defined on [0, 2]. For the partition P = {0,5, 1,,, 2} of [0,2], find S(F,P) and S(f,P). (7.5) 4. (a) Use sequential criteria for integrability to show that the function f(x) = x? is integrable on [0, 1] and find So f. (7.5) 10) Let f: → I be defined as (2) = 13, iF 15x53 otherwise . Then prove that f is .. integrable on [0, 5] and find J, 5. (7.5) (c) Let f.g : [a, b] → R be integrable functions on [a, b] . (1) If F(x) ≥ 0, for all x € [a, b], then prove that 0 ≤ Jaf. (i IF F(x) ≤ 9(x), for all x E [a, b], then prove that S"/ ≤ Jag. (7.5) 5. (a) Discuss the pointwise convergence of the following sequences of functions: (1) < In > where fn(x) = - -, XER n x'+nx (b) Define uniform convergence for a sequence of functions. Let In(x)=x™ ,* € [O, 1], then show that < In > is uniformly convergent on [0, c] where c < 1 but it is not uniformly convergent on [0; 1]. (7.5) (c) State the Cauchy Criterion for uniform convergence of a sequence of functions. Test the sequence < fn > for uniform convergence on [0, 1] where 6. (a) State Weierstrass M-Test for uniform convergence of series of functions. Test the following series for uniform convergence on R, (7.5) (b) Find the interval of convergence and the radius of convergence for the following power series, (7.5) )Show thattanx=x-.,-15x≤1. And use Abel's theorem to deduce that *=1-}+-}+... (7.5)